The distance between a naive cumulative estimator and its least concave majorant

We consider the process Λ̂n−Λn, where Λn is a cadlag step estimator for the primitive Λ of a nonincreasing function λ on [0,1], and Λ̂n is the least concave majorant of Λn. We extend the results in Kulikov and Lopuhaä (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of Λ̂n−Λn converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the Lp-distance between Λ̂n and Λn.